When to include the average of a Level 1 predictor as a Level 2 predictor? Reflecting on this, I came to the conclusion that the most obvious reason to do it was if you were interested in whether the average of the Level 1 variable in a particular Level 2 unit impacts the outcome variable, even once we've controlled for the Level 1 variable itself.
I've also seen some other reasons for doing it suggested in the literature.

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*Raudenbush & Bryk (2002) p. 261 suggest that it is one option to fix problems with bias that would otherwise occur when an omitted Level 2 predictor is associated with a Level 1 predictor.


*Gelman & Hill (2006) p. 480 also recommend doing it in an example they give in which adding a predictor would otherwise increase the residual variance.


*While not offering any comment on its merits, Heck et al (2014) give an example of a random intercepts model where individual pupils are nested in schools and both individual SES and the average SES at a pupil's school are used to predict individual test scores. The model is as follows:
$Y_{ij} = \gamma_{00} + \gamma_{01}\text{ses_mean} + \gamma_{02}\text{pro4yrc} + \gamma_{03}\text{public} + \gamma_{10}\text{ses} + u_{0j} + \epsilon_{ij}$
I checked and in their book the Level 2 variable ses_mean is calculated just from averaging the SESes of other pupils in the sample who are from the same school, i.e. it can be calculated simply from the Level 1 variable SES and a knowledge of which pupils go to which schools.
My questions:

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*Are all these three reasons valid? Are there other good reasons to include the average of a Level 1 predictor as a Level 2 predictor?

*What are some reasons to not include the average of a Level 1 predictor as a Level 2 predictor?

*Should the average of a Level 1 predictor be routinely used as a Level 2 predictor, or only in special circumstances?

Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.
Heck, R. H., Thomas, S. L., & Tabata, L. N. (2013). Multilevel and longitudinal modeling with IBM SPSS. Routledge.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). Sage.
 A: I just responded to your other question on this issue. To address your questions:

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*Are all these three reasons valid? Are there other good reasons to
include the average of a Level 1 predictor as a Level 2 predictor?

Yes, they are valid. Another reason is for understanding the decomposition of variance in the outcome. When you enter in a level 1 predictor into a model, you will notice that both the level 1 and level 2 variance can be impacted. By explicitly separating these two sources of influence through either group-mean centering or adding the level 2 mean, you better understand the nature of that variable's influence on the outcome.


*What are some reasons to not include the average of a Level 1
predictor as a Level 2 predictor?

I am not aware of any other than the potential for overfitting. If all the action is at level 1 and you include a bunch of means of the level 1 variables, then model fit criteria will likely tell you that you are not helping yourself by including these predictors.


*Should the average of a Level 1 predictor be routinely used as a
Level 2 predictor, or only in special circumstances?

I tend to think they should, but that is a personal preference. Other people might prefer to group-mean center the level 1 variables and only include level 2 variables that quantify phenomena at level 2.
